Dynamic Programming Power Control for Doubly Fed ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 5, SEPTEMBER 2008

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Dynamic Programming Power Control for Doubly Fed Induction Generators David Santos-Martin, Jose Luis Rodriguez-Amenedo, Member, IEEE, and Santiago Arnalte

Abstract—This paper presents a novel control strategy, dynamic programming power control (DPPC), to be applied to doubly fed induction generators most commonly used in wind energy applications. Although the technique can be implemented to control both rotor and grid converters, it will hereby be expounded the former, which regulates stator active and reactive powers. The results obtained are compared with those from other techniques, such as direct torque/power control (DTC/DPC) through the use of experimental tests, and indicate that DPPC achieves gains in dynamic response and considerable improvements in terms of ripple reduction and frequency spectrum as a result of constant switching frequency operation. The validation of results has been performed through experimental tests on a 6-kW generator. Index Terms—Direct power control (DPC), doubly fed induction generator, dynamic programming.

I. INTRODUCTION

D

URING the last two decades, large scale integration of wind energy has become a fact, due to social and geopolitical concerns. In electrical terms, the evolution from short-circuit induction generators and wound rotor induction generators with supersynchronous cascade, has reached its end with the introduction of doubly fed induction generators (or doubly fed asynchronous generators) with bidirectional, and partially rated, power flow inverters. This type of generator has now become the most deployed system, and competes with other synchronous and full-rated, back-to-back systems. The rapid evolution of wind energy has focused, so far, on its electrical components, generators, and power electronics. The application of known control techniques, such as flux oriented vector control, has proved until now sufficient for the accomplishment of the initial requirements. Grid operators, wind farm owners, and large scale installations of wind power are redefining these specifications and aim to increase the quality and robustness of the systems during normal and perturbated operation. Advanced control techniques are required to match these new needs. The classical control theory, based on frequency analysis and linear regulators, was first used to control induction motors and, thereafter, in wind energy applications. Vector control, through its rotational transformations, feeds cascade loops and decouples control between the electrical torque and the excitation

Manuscript received August 30, 2007; revised November 25, 2007. Current version published November 21, 2008. Recommended by Associate Editor J. Guerrero. The authors are with the Department of Electrical Engineering, University Carlos III, Madrid 28911, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2008.2001908

rotor current, which satisfied the initial needs. But its linear nature and lack of robustness when facing parameter changes and/or changes in operational conditions were its main drawbacks. Modern control techniques based on state variable analysis, i.e., pole placement, linear quadratic regulators (LQR), and robust control designs such as linear quadratic Gaussian (LQG) or , have not proved popular singular value based solutions like when applied to control motors or generators. More attention has been given to the development of estimators and observers of angle or flux variables, not always supported in the optimal estimation theory proposed by Kalman and Bucy [1]. Due to both the nonlinear nature of the inverter, of finite possible states, and the linear time-varying nature of the machine model, nonlinear control techniques, such as direct torque control (DTC) or direct power control (DPC), have been proposed in the last few decades. The basic principle of these techniques is the selection of a space vector and an averaged control signal, in order to instantaneously control during one sample of time, both the electromagnetic torque or active power and the flux magnitude or reactive power. The main advantages of direct control are its high dynamics due to the saturation control law, its performance robustness that depends solely on input error and rotor position, and its simplicity of implementation. The drawbacks, on the other hand, are directly related to the advantages. As in all controllers, a high bandwidth may result in poor noise rejection, and the tracking on undesirable high dynamics, the consequence being an important torque or power ripple. The controllers dependence on error evolution and the absence of a symmetrical commutation pattern leads to an inherently variable switching spectrum, that varies with rotor speed and operational conditions, and where special care must be taken for the eventuality of a near zero slip. Many methods have been proposed to solve these problems, and they mainly involve switching frequency imposition and ripple reduction. The variation of the switching frequency according to the amplitude of the hysteresis band and to the different operation points, has led to detailed studies that consider the following: to estimate the evolution of control variables [2]; to include more convenient commutation techniques, such as space vector modulation [3], [4]; to calculate the precise time to impose vectors [5], and even include linear regulators [6], [7]. Other proposals are focused on using multilevel converters [8]. Recently, DTC has been proposed to control a DFIG [9], [10]. Based on the principles of DTC [11], [12], another nonlinear controller, the DPC, was proposed by Onishi [13] for the instantaneous control of the active and reactive powers of three phase PWM inverters. The advantages and disadvantages are more or less equal to those of the DTC, while yielding to

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The stator and rotor flux linkage vectors can be defined for any static or rotating reference as

(2) where are the stator, rotor, and mutual inductance, respectively. The stator current can be expressed as

(3) where is the leakage factor. Yielding, when considering or neglecting the stator resistance, an exact or approximate expression

Fig. 1. Stator voltage and rotor flux linkage vectors and its projections with synchronous reference dq.

similar solutions. For switching frequency imposition at any working point in [14], it is proposed to regulate dynamically the hysteresis bandwidth, while [15] studying the DPC with space vector modulation and linear regulators. The proposal includes an estimation of variables in order to simplify or increase robustness as presented in [16] and [17], and likewise in [18], where the standard controller is modified to deal with unbalanced disturbances. DPC has also been applied to DFIG [19], [20] with a simple lookup table and optimum use of non-active vectors. In [21], special attention is given to the estimation of fluxes. This paper presents a novel approach to control doubly fed induction generators. Although the technique is valid for both types of converters, it will only expound the rotor side converter. The algorithm dynamic programming power control (DPPC) applies the Bellman [22] theory for optimal control of discretetime systems, and results in a closed loop, that is generally nonlinear, and a feedback scheme. The method defines a quadratic time-domain performance criterion or cost function, that determines the best or optimum policy from any operation point to another, according to the previously defined performance function and applying the correct converter switches sequence. II. CONCEPT OF DPC FOR DFIG In doubly fed induction generators the rotor side inverter, working as a voltage source, controls the stator active and reactive powers directly, by means of the rotor flux and applying the appropriate voltage vector. Both active and reactive powers are scalar magnitudes and act independently of the reference system

(1)

(4)

As shown in (4), considering and as constant values, the variation of the angle between the rotor flux and the stator voltage, and the variation of the rotor flux modulus will produce increments in the stator active and reactive powers (see Fig. 1). , any small variations of the angle When the angle will affect mainly the active power and, by contrast, variations in the modulus of rotor flux will affect primarily the reactive power. Considering (5), the rotor voltage may modify, within each sampling period, the angle and modulus of the rotor flux and, therefore, also the active and reactive powers in the desired way. Similar conclusions are obtained in [10]–[12]. For a twoand two nonlevel converter, six active vectors are possible active vectors (5) where is the switching period. The DPC strategy is based on these assumptions and summarizes the effects of the rotor voltage on the evolution of the stator active and reactive powers on a table, which predicts for every 60 sector the most convenient vector to apply in order to increase or decrease the power. Two hysteresis controllers, that compute signals in power error, feed the necessary input to the table. As shown in Fig. 3, the DPC strategy is defined to apply the same vector during a 60 degree sector, but this classical strategy can be improved by using a 30 sector, as the vectors that maximize the changes in the active and reactive powers are the same only for a 30 sector, due to the 30 displacement between power increment patterns.

SANTOS-MARTIN et al.: DYNAMIC PROGRAMMING POWER CONTROL FOR DOUBLY FED INDUCTION GENERATORS

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III. DYNAMIC PROGRAMMING POWER CONTROL (DPPC) A. General Concept Direct torque and DPC have been recently proposed as suitable techniques for doubly fed induction generators and two level inverters. Several authors have addressed approaches to improve their behavior by reducing the torque ripple and, therefore, guaranteeing a constant switching frequency. Most of these studies are focused on the direct torque controlled induction machine. In a general case, the approximated dynamic behavior of a system in discrete state space notation takes the form (6) Fig. 2. State evolution that minimizes the performance criterion.

where is the process state, the control input, and the sampling period. As in conventional optimum control techniques, the control input may be chosen to minimize a specified performance criterion (7) where functions and define the dynamic behavior and the being a linear, even, cost of control. In the particular case of and invariant function, and a quadratic performance criteria, the optimal control law becomes a linear function, whereby gain matrix will be obtained by solving the matrix Riccati equation [1] (8) Determining the optimum control law for non-quadratic performance criteria or nonlinear process, must be approached through dynamic programming or, perhaps, calculus of variations. The algorithm DPPC applies the Bellman [22] theory for optimal control of discrete-time systems, which considers an optimal policy as the succession of optimal decisions. to by choosing Fig. 2 illustrates how to evolve state the path that minimizes the overall performance criteria, calculated by adding the cost-to-go for each stage. It is assumed that can evolve to only a finite number of states. It is important to mention that choosing the path that minimizes the criteria stage by stage does not guarantee the overall optimum path. For example, if evolving from A to C has a lower cost than doing so from A to B; this one-step strategy is called suboptimal. In an infinite horizon control case with nonlinear or perturbed models and control constraints, in order to keep the state of a system near a desired point, it is convenient to use multistage lookahead policies combined with rollout algorithms. At each stage, the optimal control problem is solved over a fixed length horizon, starting from the current stage. The first component of the corresponding optimal policy is then used as the control factor of the current stage, while the remaining components are discarded. The process is then repeated at the next stage once it is revealed. This lookahead and rolling horizon approach is typically not optimal, but it is high performance and has a low

computation cost. The simplest possibility is to use a one-step lookahead policy. This strategy is also valid for tracking reference control problems. B. Discrete Dynamic Model for DFIG As shown previously, the approximated dynamic behavior of a system is needed in order to apply the DPPC strategy to the DFIG. The discrete state space notation is chosen to represent machine model referred to in the synchronous reference the

(9) is the process state vector conwhere taining the stator and rotor currents, the input vector containing the stator and rotor voltages, and is the sampling period. is a time varying matrix that depends on rotor speed, although it can also be considered constant within short periods of time due to high rotor inertia; and is a constant matrix in the case of neglecting saturations, as is C (see Appendix for matrix definitions). contains the instantaneous The output vector stator active and reactive powers, and is considered a constant matrix. Thus, using (9), it can be demonstrated that the increment equation for the output vector is

(10)

The power increment is, for a short period of time, a linear function with a slope that depends on the applied rotor voltage, and . This result is only valid when considering average values within the discrete time. As is well known, a two-level inverter provides eight possible switching states, made up of six and two zero switching states. active Operating with the rotating transformation and the conversions gained from switching states to voltages, the rotor voltage

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Fig. 3. Simulated reactive power possible increments and an example of switching selection for slip angle variation within a fixed period and synchronous reference.

trajectory when eliminating the sine functions, and therefore, angle or time

Fig. 4. Power evolution and possible increments for all available control vectors with constant vector during the whole sampling period.

can be expressed by means of vectors, resulting in an instantaneous power increment as follows:

(13) However, not every point of the circumference provides a possible increment at any time. As seen in Fig. 5(a), only seven power increments are possible for a given rotor position, and the seven possibilities will rotate while the slip angle changes. When applying three vectors to the same switching period, the average power increment is given by the gravity center between the two possible increments, their weights corresponding to their active times, as seen in Fig. 5(b) and (c). As shown in these figures, and due to the non-stationary nature of the system, it is not possible to apply zero power increments at any time, so even when reaching the reference point it will always be necessary to move away. C. Performance Criterion

when

applied

(11)

when (12) where is the index for one of the six active vectors, the the dc bus link voltage level, the stator rotor slip angle, , the machine leakage inductances. voltage, and Equations (11) and (12) show the instantaneous linear power variation within a switching period, with a slope that depends and . There is only a on the applied rotor voltage , finite number of possible slopes, seven, that depend on the rotor slip angle (see Fig. 3) describing sinusoidal values, when using the synchronous reference. How the choice of vector will determine the slopes for both active and reactive powers must be considered when choosing a priority strategy (see Fig. 4). It is very enlightening to formulate and draw the state trajectories of the system to fully understand how it works. As shown in (13), the active and reactive power increments have a circular

The control law could be solely defined to match stability and dynamic specifications, but defining a performance criterion allows designing a control system that controls as much as can be controlled. In our case, the use of this criterion also allows calculating the control law that simultaneously evolves the active and reactive powers while sharing priorities. The choice of a quadratic law is a practical compromise between formulating the real control problem and an artificial problem of easier solution

(14) is the error of the output where is the control vector containing the vector, and are the semi-definite stator and rotor voltages, and positive matrix that share the importance of regulating active and reactive errors, and therefore, its ripple and dynamics may change, for example when tracking a reference or mainis a varying matrix that may favor the taining the state. use of some vectors, in order to reduce the inverter number of switching or when performing a more constant switching frequency.


Fig. 5. Rotating diagrams in synchronous reference with possible power increments when applying for one sampling period: (a) V V V vectors.

Fig. 6. Schematic active power evolution for one possible control combination (V i(k ); V i(k ); V i(k )) at a chosen time t.

Finally, it must be defined the number of stages to include in the optimization process; whether to use a constant, or several vectors within a switching period; and to compute the optimal policy, and, thus, the control law that minimizes the performance criterion.

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V

vector; (b)

V V V

; (c)

Fig. 7. Schematic diagram of the proposed DPPC algorithm.

applying only and for correspond to applying the power increment related to the middle point between and [see Fig. 5(c)]. The resulting performance criteria for the three cases are

D. Case Study of DPPC DPPC is exemplified with the use of three cases. In the first , a constant vector is applied during the switching one period and only a one-step lookahead policy [see Figs. 4 and , a constant vector 5(a)] is selected. In the second case, is used during the switching period, and a three-step lookahead , policy of rolling horizon is selected. In the third one, only one stage is chosen and three vectors applied during the switching period, which results in a subset of promising controls by dividing the one-step time in a discrete grid, where three vectors can be applied [see Fig. 5(b) and (c) and Fig. 6]. If the applied vector is not constant during a single switching period, the average slope for this period will correspond to the average value of the different used vectors, which allows for any slope between the maximum and minimum within that operating point [see Fig. 5(b) and (c)]. for any of the six possible In the case of applying , the choice of the sharing time will combinations . This paramdetermine the applied power increment eter can be understood as the weight of the central vector . For example, the average power increments for correspond to applying only , for correspond to

(15)

For and , the calculation of the vectors to be and are minimized. applied is done so , the calculation effort to obtain the optimum For is higher. The continuous variable combination can be discretized in a finite set of values (see Fig. 6) calculated and all possible combinations of varying from 1 to 6 and from 0 to . The combination to be applied at each switching period is the one that minimizes . The control scheme illustrated in Fig. 7 undertakes the following tasks: it estimates the active and reactive power evolution for the selected number of stages, calculates the cost-to-go for every possible path between the stages, as well as the optimum policy between the present power and the

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Fig. 9. Apparent power ripple: DPC (100 s, O), DPC (50 s, (100 s, ).

1

),

DPPC

Fig. 8. Schematic diagram of the test bench.

reference values within the selected number of stages, and applies the resulting control law that minimizes the performance criterion. The cases presented in this paper use the same weighting matrix of the performance criterion Fig. 10. Stator currents: DPC (100 s) high ripple, DPC (50 s) medium ripple, (100 s) low ripple.

(16)

DPPC

where to balance the importance of active and reactive power variations. Because of the positive definiteness of and , this technique guarantees that active and reactive power tracking errors are bounded, thus, procuring a stable feedback system. In order for such a controller to exist, it is sufficient to have a positive number of stages, so as that at any initial state a sequence of controls can be found that derive in a final state within a desired tolerance, while satisfying the control constraints. This rolling horizon policy is typically suboptimal, but highly performing. IV. EXPERIMENTAL RESULTS To validate the proposed control strategy, DPPC, a number of experimental results on a 6-kW test bench were carried out. This strategy has been compared with DPC, with and without the use of non active vectors. The test bench is composed of a 6-kW DFIG driven by a dc motor emulating a wind turbine. The system is controlled by a DS1006 control board, and reaches 50 s of cycle time. The dc . The ac line voltage is supplied by link level is set to 550 the laboratory grid at 50 Hz and 380 V. The rotational speed is set at 800 r/min, which varies slightly during high changes in reference of active power. A differential encoder with 5000 pulses/rev is used to obtain the rotor angle (see Fig. 8). A comparison between the different control techniques has been achieved by calculating the total harmonic distortion (THD) (see Figs. 9 and 13), active and reactive power ripple (see Fig. 14), dynamic response (see Figs. 11 and 15), and the frequency spectrum (see Figs. 12 and 16).

Fig. 11. Stator currents (THD): DPC (100 s, O), DPC (50 s, (100 s, ).

1

),

DPPC

A. Power Ripple The active and reactive power ripple values depend on whether the DPC or the DPPC strategy is applied. shows better results for both measurements at any working point (see Fig. 9). These results are easily explained if it is considered that the DPC strategy, a simple and robust algorithm, has no optimization criteria, taking into account only sector consideration, whereas angle position is not used. This does not , which utilizes the knowledge of the occur with the plant to control as efficiently as can be done at any given time, attending to the designed performance criterion that includes minimizing the active and reactive power ripple. Each strategy shows nearly the same ripple for all working points.


Fig. 14. Rotor voltage and the underlying first harmonic for DPC,

DPPC , DPPC .

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DPPC ,

Fig. 12. Stator active power during reference change for DPC (50 s).

Fig. 15. Frequency spectrum in 10 kHz for rotor voltage and stator current, respectively, with DPC (50 s).

Fig. 13. Stator power during reference change for

DPPC

(100 s).

TABLE I EXPERIMENTAL RESULTS

Fig. 16. Frequency spectrum in 10 kHz for rotor voltage and stator current, respectively with (100 s).

DPPC

B. Total Harmonic Distortion (THD) shows much better results of the stator curAgain rent THD (see Figs. 10 and 11), as a consequence of optimizing the power ripple. The THD ratio between different strategies is nearly independent of the amount of injected current. C. Dynamic Response As shown in Figs. 12 and 13 and Table I, both nonlinear controllers are extremely dynamic, with time responses near 1 ms, and provide results that are sufficient for standard operational conditions and highly convenient when severe perturbations occur.

D. Frequency Spectrum One of the main advantages of compared to DPC, is related to the frequency spectrum (see Figs. 14–16). The broad spectrum for voltages and currents when applying DPC strategy is due to the use of hysteresis regulators; in contrast, guarantees the existence of the same number of commutations by switching period (see Fig. 6) obtaining a constant switching frequency operation for the rotor voltage and a very narrow spectrum for the stator current. The number of switching and, consequently, the commutation losses remain similar in both cases, DPC (50 s) and

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Fig. 19. Sensibility study with power ripple Ps function weight) for (100 s).

DPPC

(1), Qs ( ) versus

(cost

Fig. 17. Sensibility study with power ripple versus error in all parameters for (100 s) (Ps , Qs ) and DPC (50 s) Ps and, Qs .

DPPC

(1)

( )

()

Fig. 20. Sensibility study with stator current THD versus (100 s). weight) for

DPPC

Fig. 18. Sensibility study with stator current THD vs error in all parameters (100 s) . , and DPC (50 s)

DPPC

(1)

()

(100 s), though those of the DPPC can be reduced, by enlarging the switching periods while sacrificing some of the excellent THD and ripple results. E. Robustness The robustness of the different strategies is analyzed by performing a sensibility study. This sensibility error study has been carried out by incorporating a 50% error in resistor and inductance parameters within the estimation model of the strategy (see Figs. 17 and 18). The system is always stable and the results show an improvement over those of the DPC (50 s). (see Figs. 19 Finally, the influence of cost function weight and 20) is considered, and the range of variation of this weight is established for achiving efficient THD and ripple levels at (0.3–0.7).

(cost function

For the DPPC algorithm expounded in this paper, the one-step shows the best results (see and three-vector strategy Table I), due to the availability of a higher number of possible combinations (see Fig. 5). However, an increase on computation requirements would be needed for a very fine time grid or for multiple-step strategies, being necessary in some cases to enlarge the switching period. A compromise between complexity and computation capacities is required to optimize results. V. CONCLUSION The proposed algorithm is thus capable of controlling the decoupled active and reactive powers of doubly fed induction generators, and provides excellent dynamic response, with minimum ripple and concentrated frequency spectrum, as demonstrated by the experimental results. The results have been favorably compared with those of classical direct power control as offering a higher performance. The sensibility study ratifies the robustness of the proposed method. All these factors make DPPC an attractive proposition that ensures a high performance


control of drives for DFIG generators, and other types of generators used in wind energy applications. APPENDIX The equation shown at the bottom of the previous page, where , , and are the stator, rotor, and serial iron resistance, and are the stator and rotor inductance; respectively; the magnetizing inductance; the synchronous speed; the the leakage factor slip and

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[14] S. Chen and G. Joós, “Direct power control of active filters with averaged switching frequency regulation,” in Proc. IEEE 35th Power Electron. Specialists Conf. (PESC), Jun. 2004, vol. 2, pp. 1187–1194. [15] M. Malinowski and M. Jasinski, “Simple direct power control of three-phase PWM rectifier using space vector modulation (DPC-SVM),” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 447–454, Apr. 2004. [16] T. Noguchi, H. Tomiki, S. Kondo, and I. Takahshi, “Direct power control of PWM converter without power source voltage sensors,” IEEE Trans. Ind. Appl., vol. 34, no. 3, pp. 473–479, May–Jun. 1998. [17] M. Malinowski, M. P. Kazmierkowski, S. Hansen, F. Blaabjerg, and D. Marques, “Virtual flux based direct power control of three-phase PWM rectifiers,” IEEE Trans. Ind. Appl., vol. 37, no. 4, pp. 1019–1027, Jul./Aug. 2001. [18] G. Escobar, A. M. Stankovic, J. M. Carrasco, E. Galvan, and R. Ortega, “Analysis and design of direct power control (DPC) for a three phase synchronous rectifier via output regulation subspaces,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 823–830, May 2003. [19] R. Datta and V. T. Ranganathan, “Direct power control of grid-connected wound rotor induction machine without rotor position sensors,” IEEE Trans. Ind. Appl. Power Electron., vol. 16, no. 3, pp. 1076–1082, May 2001. [20] S. Arnalte and J. L. Rodriguez-Amenedo, “Sensorless direct power control of a doubly fed induction generator for variable speed wind turbines,” in Proc. IEEE 10th Power Electron. Motion Control Conf. (EPE-PEMC), Sep. 2002, vol. 1, pp. 1–12. [21] L. Xu and P. Cartwright, “Direct active and reactive power control of DFIG for wind energy generation,” IEEE Trans. Energy Convers., vol. 21, no. 3, pp. 750–758, Sep. 2006. [22] R. Bellman and S. Dreyfus, Applied Dynamic Programming. Princeton, NJ: Princeton Univ. Press, 1962. David Santos-Martin received the B.Sc. degree in electrical and electronic engineering from E.T.S. Industrial Engineering of Madrid (ETSII-UPM), Madrid, Spain, in 1997, the M.S. degree in control engineering from The École Supérieure dÉlectricité (SUPÉLEC-Paris), Paris, France, and the Ph.D. degree in electrical engineering from the University Carlos III, Madrid, Spain. Currently, he is an Assistant Lecturer with the Department of Electrical Engineering, University Carlos III of Madrid. Prior to this, he was with Iberdrola from 2001 to 2007 and with Ecotecnia-Alsthom from 2000 to 2001. His research interests include power electronics, application of power electronics to power systems, and advanced control techniques applied to renewable energy.

Jose Luis Rodriguez-Amenedo (M’01) received the B.S. degree in energy engineering from the Universidad Politécnica de Madrid, Madrid, Spain, in 1993, and the Ph.D. degree in electrical engineering from Universidad Carlos III de Madrid, Madrid, Spain, in 2000. He is currently an Assistant Professor with the Universidad Carlos III de Madrid. His current research interests include control drives and wind energy systems.

Santiago Arnaltes received the Ph.D. degree in electrical engineering from Polytechnic University of Madrid, Madrid, Spain, in 1993. Since 1997, he has been an Associate Professor with the Department of Electrical Engineering, Carlos III University of Madrid, Madrid, Spain. His main research interests include grid integration of wind energy and control of electrical drives and FACTS, mainly for wind energy applications.